Subgroups — #11758561

Order 992.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊D₃₁	C₂	1	4	yes	maximal, complemented
3	#92497	C₂	1	4	yes	maximal, complemented
4	D₄×D₃₁	C₂	1	4	yes	maximal, complemented
5	#92495	C₂	1	2	yes	maximal, complemented
6	C₂³×C₃₁	C₂²	1	2	yes	complemented
7	C₂²×D₃₁	C₂²	1	4	yes	complemented
8	D₄×C₃₁	C₂²	1	4	yes	complemented
9	C₂×C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
10	C₂²⋊D₃₁	C₂²	1	4	yes	complemented
11	C₂²⋊D₃₁	C₂²	1	8	yes	complemented
12	C₂×Dic₃₁	C₂²	1	4	yes	complemented
13	D₁₂₄	C₂²	1	2	yes	complemented
14	C₄×D₃₁	C₂²	1	4	yes	complemented
15	Dic₆₂	C₂²	1	2	yes	complemented
16	C₂²×C₃₁	—	4	2	no	—
17	C₂²×C₃₁	C₂³	1	4	yes	—
18	C₂²×C₃₁	C₂³	1	1	yes	characteristic
19	C₂×D₃₁	—	4	4	no	—
20	C₂×D₃₁	C₂³	1	4	yes	—
21	C₄×C₃₁	C₂³	1	2	yes	—
22	Dic₃₁	C₂³	1	4	yes	—
23	C₂×C₃₁	—	2	4	no	—
24	C₂×C₃₁	—	2	1	no	—
25	C₂×C₃₁	C₂⁴	1	1	yes	characteristic
26	D₃₁	—	2	4	no	—
27	H(5,𝔽₂)	—	31	1	no	maximal, Sylow
28	C₃₁	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
29	#41	—	31	4	no	—
30	#41	—	31	2	no	—
31	C₂×D₄	—	31	4	no	—
32	C₂×D₄	—	31	4	no	—
33	C₂×D₄	D₃₁	1	1	yes	characteristic, complemented
34	C₂³	—	31	4	no	—
35	C₂³	C₂×D₃₁	1	2	yes	complemented
36	Q₈	—	31	2	no	—
37	D₄	—	31	8	no	—
38	D₄	—	31	4	no	—
39	D₄	—	31	2	no	—
40	D₄	C₂×D₃₁	1	4	yes	complemented
41	C₂×C₄	—	31	4	no	—
42	C₂×C₄	—	31	4	no	—
43	C₂×C₄	C₂×D₃₁	1	1	yes	characteristic, complemented
44	C₂²	—	4	2	no	—
45	C₂²	—	31	4	no	—
46	C₂²	—	124	4	no	—
47	C₂²	C₂²×D₃₁	1	4	yes	—
48	C₂²	C₂²×D₃₁	1	1	yes	characteristic
49	C₄	—	31	4	no	—
50	C₄	C₂²×D₃₁	1	2	yes	—
51	C₂	—	2	1	no	—
52	C₂	—	2	4	no	—
53	C₂	—	62	4	no	—
54	C₂	C₂³×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₃₁
commutator, socle, cosocle	C₂×C₃₁	C₂⁴
Fitting	C₂×D₄×C₃₁	C₂
hypercenter	C₂×D₄	D₃₁
hypocenter	C₃₁	H(5,𝔽₂)
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ C₂×D₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×D₄×C₃₁	C₂	semidirect
2. C₂×C₂²⋊D₃₁	C₂	semidirect
3. #92497	C₂	semidirect
4. D₄×D₃₁	C₂	semidirect
5. #92495	C₂	semidirect
6. C₂³×C₃₁	C₂²	semidirect
7. C₂²×D₃₁	C₂²	semidirect
8. D₄×C₃₁	C₂²	semidirect
9. C₂×C₄×C₃₁	C₂²	semidirect
10. C₂²⋊D₃₁	C₂²	semidirect
11. C₂²⋊D₃₁	C₂²	semidirect
12. C₂×Dic₃₁	C₂²	semidirect
13. D₁₂₄	C₂²	semidirect
14. C₄×D₃₁	C₂²	semidirect
15. Dic₆₂	C₂²	semidirect
28. C₃₁	H(5,𝔽₂)	semidirect
33. C₂×D₄	D₃₁	semidirect
35. C₂³	C₂×D₃₁	semidirect
40. D₄	C₂×D₃₁	semidirect
43. C₂×C₄	C₂×D₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×D₄×C₃₁	2
C₂×C₂²⋊D₃₁	2
#92497	2
D₄×D₃₁	2
#92495	2
H(5,𝔽₂)	31